Large Deviation Principles

Abstract

This chapter is devoted to the study of large deviation principles and equidistribution results for periodic points and iterated preimages of expanding Thurston maps without periodic critical points. The idea is to apply a general framework devised by Y. Kifer [Ki90] to obtain level-2 large deviation principles, and to derive the equidistribution results as consequences. In Sect. 7.1, we give a brief review of level-2 large deviation principles in our context. We then establish some characterization of topological pressure in our context in Sect. 7.2 before providing a proof of Theorem 7.1, establishing level-2 large deviation principles in the context of expanding Thurston maps without periodic critical points and given Holder continuous potentials. Finally in Sect. 7.4, we derive equidistribution of periodic points and iterated preimages with respect to the equilibrium state in our context with fairly flexible choices of weight at each point.